Maybe you get a better understanding if you remember the automata generating these languages.
Regular languages are generated by regular automata. They have only have a finit knowledge of the past (their compute memory has limits) so everytime you have a language with suffixes depending on prefixes (palindrome language) this can not be done with regular languages.
Context-free languages are generated by nondeterministic pushdown automata. They have a kind of knowledge of the past (the stack, which is not limited in contrast to regular automata) but a stack can only be viewed from top so you don’t have complete knowledge of the past.
Context-sensitive languages are generated by linear-bound non-deterministic turing machines. They know the past and can deal with different contexts because they are non-deterministic and can access all the past at every time.
Unrestricted languages are generated by Turing machines. According to the Church-Turing-Thesis turing machines are able to calculate everything you can imagine (which means everything decidable).